Comparison of Nested and Non-nested Direct Solver Performance for Generalized Source Integral Equations

Fast direct integral equation (IE) solvers rely on the initial computation of a hierarchically-compressed blocked representation of the method of moments moment matrix, which is then factorized efficiently using specialized arithmetic [1]. The hierarchical partitioning of the matrix is inherited form that of the basis and testing functions defined on the computation domain. The blocks deemed "admissible" for compression are typically identified using a distance- and size-based criterion and are conventionally expressed by their low-rank (LR) approximation. Direct solvers can be classified by the type ("weak" or "strong") of admissibility criterion in use and by whether the bases for the LR approximations of blocks are nested or not. matrix [1] and-matrix [2] methods typically make use of a "strong" admissibility criterion, requiring strict separation between the subdomains on which the basis and testing functions participating in the compressed interaction are defined. While in the matrix approach each block is compressed independently, in the-matrix compression, the bases used for computing LR approximation of blocks associated with parent basis and testing function clusters are combinations of already reduced bases associated with their child subclusters, in what can be viewed as an algebraic variation (with a tighter bound on the rank) of the multilevel fast multipole algorithm [3]. The differences in performance between the two schemes and associated direct solvers have been studied extensively for conventional IEs, which make use of the free-space Green's function kernel. Recently, with the aim of increasing the LR-compressibility of moment matrices, for impenetrable essentially convex scatterers, generalized source integral equation (GSIE) formulations have been proposed [4], [5]. These IE formulations make use of modified kernels that exhibits significant attenuation of the radiation by elemental sources into the volume occupied by the scatterer. This reduces the effective dimensionality of the truncated (via LR approximation) interactions, which are of diminished broadside and pronounced end-fire nature. These directional properties of the GSIE wave interactions may influence the interplay between child- and parent-level bases in the matrix scheme and call for a designated comparative compressibility and performance study using the modified Green's functions.